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Pfister and Sullivan proved that if a topological dynamical system $(X,T)$satisfies almost product property and uniform separation property, then foreach nonempty compact %convex subset $K$ of invariant measures, the entropy ofsaturated set $G_{K}$ satisfies \begin{equation}\label{Bowen's topologicalentropy}  h_{top}^{B}(T,G_{K})=\inf\{h(T,\mu):\mu\in K\}, \end{equation} where$h_{top}^{B}(T,G_{K})$ is Bowen's topological entropy of $T$ on $G_{K}$, and$h(T,\mu)$ is the Kolmogorov-Sinai entropy of $\mu$. In this paper, weinvestigate topological complexity of $G_{K}$ by replacing Bowen's topologicalentropy with upper capacity entropy and packing entropy and obtain thefollowing formulas: \begin{equation*}  h_{top}^{UC}(T,G_{K})=h_{top}(T,X)\ \mathrm{and}\h_{top}^{P}(T,G_{K})=\sup\{h(T,\mu):\mu\in K\}, \end{equation*} where$h_{top}^{UC}(T,G_{K})$ is the upper capacity entropy of $T$ on $G_{K}$ and$h_{top}^{P}(T,G_{K})$ is the packing entropy of $T$ on $G_{K}.$ In the proofof these two formulas, uniform separation property is unnecessary.

Topological structures on saturated sets, optimal orbits and equilibrium states